Math 160: Study Guide - Chapters 5&6

1. A feasible region is sketched and the equations of each of the edges is given. Determine where the maximum (or minimum) value of the objective function will occur. This question is about interpreting the slope of the objective function in relation to the slopes of the edges. Five parts.
2. Construct a mathematical model in the form of a linear programming problem. Be sure to define your variables. Do not write the system or solve the problem.
3. A system is given. The non-basic variables are set equal to zero and you need to find values of the basic variables. Some of the points are completely finished to reduce the amount of time it takes while still seeing if you know what you're doing. Determine whether or not each point is feasible.
4. You are given a geometric representation of a system and some tableaus. Identify which point corresponds to the tableau and indicate where you would pivot to move to a specified point.
5. Form the dual problem, but do not solve it. If there are steps you can do in your head, you are allowed. It's the final answer that will be graded.
6. The initial tableau (after pivoting on all artificial variables in the preliminary tableau) for a non-standard maximization problem is given. Identify each variable as being basic or non-basic and give its value. Determine where the next pivot should occur.
7. Take the non-standard maximization problem and write the preliminary tableau. If you can do this without writing the system, you may do so. Do not solve the problem.
8. A tableau from a standard maximization problem is given. Identify each variable as being basic or non-basic and give its value. Determine whether or not the point is feasible. Identify (multiple choice) which objective function matches the tableau.
9. A mostly-complete tableau is given. Finish the tableau so that the tableau has the indicated outcome (the optimal solution is found, an additional pivot is required, or there is no optimal solution). There are many correct answers; you just have to give one of them. Three parts.
10. The final tableau from a dual problem is given. Identify which variables are basic and non-basic and give their values for both the dual (maximization) and original (minimization) problems.
11. A tableau from a standard maximization problem is given. Identify each variable as being basic or non-basic and give its value. Label the tableau with the following items: the variable that is basic in each row, the pivot column, the appropriate ratios, the pivot row, the pivot element, the entering variable, and the exiting variable. Identify the increase in the objective function after the next pivot.
12. A linear programming problem is presented with the constraint information in table form. The final tableau is given. Assuming a normal setup, identify the values of each of the variables and give the interpretation of that value in the context of the original problem. This will involve interpreting slack and or surplus variables as well as the decision variables and objective function.
13. Identify whether each problem is a standard maximization problem, is a standard minimization problem, can be changed into a standard maximization problem (explain how), can be changed into a standard minimization problem (explain how), or is a non-standard problem that must be worked using the Big-M method. In the cases where the problem can be changed into a standard problem, indicate how. For example, you might write "Standard minimization after multiplying the second constraint by -1." Six parts.

Notes

• There is a take home exam that is due the day of the in-class exam. It is worth 30 points. See below for those problems.
• The in-class exam is worth 95 points.
• The in-class exam is designed to be worked without the simplex or pivot programs on the calculator. There are no tableaus that need entered into the calculator. The only place you might need a calculator is for solving a system of equations or finding the ratios for some of the tableaus.
• There are no difficult or time consuming questions on the exam. It is a very conceptual exam. Either you know it or you don't.
• Know the difference between a problem, system, and tableau. Also know the difference between a preliminary tableau and an initial tableau for a non-standard problem. Problems ask for a specific type of answer.
• Always make sure that the cleared column has a 1 as its sole number before trying to interpret the values in a tableau. For example, if the coefficient in the x₂ column is 3 and the right hand side is 12, then the equation is 3x₂ = 12 and the value of x₂ is 4, not 12.

Points per problem

Take Home Exam Problems

Work the following even problems from the textbook. You are free to use the simplex and/or pivot programs once the tableaus are set up. You should define your variables, write the initial problem, the initial system, the initial and final tableaus, as well as the answer to the question. You should write the intermediate tableaus only if you are using the pivot program but not if you are using the simplex program. Each question is worth 6 points.

For story problems, interpret the meaning of all of the decision, slack, and surplus variables, not just the decision variables.

1. 6.2.44. Write the initial problem, initial system, and then use the simplex program
2. 6.3.46. Write the initial problem and then form the dual problem. Write the initial system and then use the simplex program. Be sure you answer the original question.
3. 6.4.28. Write the initial system and initial tableau. Then use the pivot program and show all of the the intermediate tableaus.
4. 6.4.40, Setup and solve the problem. Hint: When properly setup, this problem can be turned into a standard maximization problem and then the simplex program ran. That's why I want you to go ahead and solve the problem, even though the instructions in the book say only to set it up. Another hint: remember that the amount loaned doesn't have to be all $3 million, so when it says things like "50% of the amount loaned", don't use 50% of $3 million, use 0.50( x₁+x₂+x₃+x₄ ).
5. Take the problem below and work it like it was a problem in section 6.1 Set up a system of equations like those in section 6.1 where you add a slack variable for each ≤ constraint. Be sure to include the non-negativity constraints when you write your system. Find the following:
   1. The number of decision variables.
   2. The number of slack variables.
   3. The number of basic variables.
   4. The number of non-basic variables.
   5. Set up a table, like we did in section 6.1, with a column for each variable, a column labeled "feasible?", and a column for the objective function P.
      1. List all of possible combinations that can be found by setting the non-basic variables equal to zero.
      2. Find the values of the basic variables for each point. If the values are "nice" decimals, you can leave them in decimal form, otherwise write them as fractions.
      3. Identify whether or not each point is feasible.
      4. Find the value of the objective function for each of the feasible solutions.
   6. What is the maximum value of the objective function and where does it occur?

Linear Programming Problem to use for #5

The take home exam is due the day of the in-class exam.